Chapter 7 Expanding brackets and factorising This chapter will show you how to ✔ expand and simplify expressions with brackets ✔ solve equations and inequalities involving brackets ✔ factorise by removing a common factor ✔ expand two brackets 7.1 Expanding brackets You will need to know how to ● multiply positive and negative numbers ● add and subtract negative numbers ● collect like terms When multiplying algebraic terms remember that x × 3 5 3 × x 5 3x y × y 5 y2 gh 5 g × h More complicated multiplications can also be simplified. EXAMPLE 1 Simplify 3f × 4g. To multiply algebraic terms, 3f × 4g 5 3 × f × 4 × g multiply the numbers then multiply 5 3 × 4 × f × g the letters. 5 12 × fg 5 12fg Multiplying a bracket You can work out 6 × 34 by thinking of 34 as 30 1 4. 6 × 34 5 6 × (30 1 4) 30 4 5 6 × 30 1 6 × 4 5 1 180 24 6 6 × 30 6 × 4 5 204 Algebra 101 M07_CME_SB_IGCSE_6867_U07.indd 101 28/8/09 11:35:25 Brackets are often used in algebra. 6(x 1 4) means 6 × (x 1 4) As in the 6 × 34 example, you have to multiply each term inside the brackets by 6. 6(x 1 4) 5 6 × x 1 6 × 4 5 6x 1 24 It is like working out the area of a rectangle that has length x 1 4 and width 6 x 4 area = 6 × x area = 6 × 4 6 = 6x = 24 Total area 5 6(x 1 4) 5 6 × x 1 6 × 4 You find the total area by adding 5 6x 1 24 the area of the two smaller When you do this it is called expanding the brackets. rectangles. It is also known as removing the brackets or multiplying out the brackets. When you remove the brackets you must multiply each term inside the brackets by the term outside the bracket. EXAMPLE 2 Simplify these by multiplying out the brackets. (a) 5(a 1 6) (b) 2(x 2 8) (c) 3(2c 2 d) You must multiply each term inside (a) 5(a 1 6) 5 5 × a 1 5 × 6 the bracket by the term outside the 5 5a 1 30 bracket. (b) 2(x 2 8) 5 2 × x 2 2 × 8 5 2x 2 16 3 × 2c 5 3 × 2 × c 5 6 × c 5 6c (c) 3(2c 2 d) 5 3 × 2c 2 3 × d 5 6c 2 3d 102 Algebra M07_CME_SB_IGCSE_6867_U07.indd 102 28/8/09 11:35:26 Expanding brackets and factorising EXERCISE 7A 1 Simplify these expressions. (a) 2 × 5k (b) 3 × 6b (c) 4a × 5 (d) 3a × 2b (e) 4c × 3d (f) x × 5y 2 Expand the brackets to find the value of these expressions. Remember, you must multiply each Check your answers by working out the brackets first. term inside the brackets by the term (a) 2(50 1 7) (b) 5(40 1 6) (c) 6(70 1 3) outside the bracket. (d) 3(40 2 2) (e) 7(50 2 4) (f) 8(40 2 3) A common mistake is to forget to multiply the second term. 3 Remove the brackets from these. (a) 5(p 1 6) (b) 3(x 1 y) (c) 4(u 1 v 1 w) (d) 2(y 2 8) (e) 7(9 2 z) (f) 8(a 2 b 1 6) 4 Expand the brackets in these expressions. Remember (a) 3(2c 1 6) (b) 5(4t 1 3) (c) 2(5p 1 q) 3 × 2c 5 3 × 2 × c 5 6c (d) 3(2a 2 b) (e) 6(3c 2 2d) (f) 7(2x 1 y 2 3) (g) 6(3a 2 4b 1 c) (h) 2(x2 1 3x 1 2) (i) 4(y2 2 3y 2 10) 5 Write down the 6 pairs of cards which show equivalent expressions. 4(x 1 2y) 4x 1 2y 2(4x 1 y) 4(2x 2 y) A B C D 8x 2 8y 4x 1 8y 8(x 2 y) 2x 2 8y E F G H 8x 1 2y 2(x 2 4y) 2(2x 1 y) 8x 2 4y I J K L You can use the same method for expressions that have an algebraic term instead of a number term outside the bracket. EXAMPLE 3 Expand the brackets in these expressions. (a) a(a 1 4) (b) x(2x 2 y) (c) 3t(t2 1 1) (a) a(a 1 4) 5 a × a 1 a × 4 Remember a × a 5 a2 5 a2 1 4a (b) x(2x 2 y) 5 x × 2x 2 x × y x × 2x 5 x × 2 × x 5 2 × x × x 5 2x2 5 2x2 2 xy Remember x × y 5 xy (c) 3t(t2 1 1) 5 3t × t2 1 3t × 1 3 5 3t 1 3t 3t × t2 5 3 × t × t × t 5 3t3 Algebra 103 M07_CME_SB_IGCSE_6867_U07.indd 103 28/8/09 11:35:27 EXERCISE 7B Expand the brackets in these expressions. 1 b(b 1 4) 2 a(5 1 a) 3 k(k 2 6) 4 m(9 2 m) 5 a(2a 1 3) 6 g(4g 1 1) 7 p(2p 1 q) 8 t(t 1 5w) 9 m(m 1 3n) 10 x(2x 2 y) 11 r(4r 2 t) 12 a(a 2 4b) 13 2t(t 1 5) 14 3x(x 2 8) 15 5k(k 1 l) Remember 3x × 4x 5 4 × 3 × x × x 16 3a(2a 1 4) 17 2g(4g 1 h) 18 5p(3p 2 2q) 5 12x2 19 3x(2y 1 5z) 20 r(r2 1 1) 21 a(a2 1 3) 22 t(t2 –7) 23 2p(p2 1 3q) 24 4x(x2 1 x) Adding and subtracting expressions with brackets Adding To add expressions with brackets, expand the brackets first, then collect like terms to simplify your answer. Collecting like terms means adding all the terms in x, all the terms in y and so on. EXAMPLE 4 Expand then simplify these expressions. (a) 3(a 1 4) 1 2a 1 10 (b) 3(2x 1 5) 1 2(x 2 4) Expand the brackets first. Then (a) 3(a 1 4) 1 2a 1 10 5 3a 1 12 1 2a 1 10 collect like terms. 5 3a 1 2a 1 12 1 10 5 5a 1 22 Expand both sets of brackets first. (b) 3(2x 1 5) 1 2(x 2 4) 5 6x 1 15 1 2x 2 8 5 6x 1 2x 1 15 2 8 5 8x 1 7 Subtracting Multiplying If you have an expression like 23(2x 2 5), multiply both terms in the brackets by 23. 1 × 1 1 23 × 2x 5 26x and 23 × 25 5 15 1 × 2 2 So 23(2x 2 5) 5 23 × 2x 1 23 × 25 2 × 1 2 5 26x 1 15 2 × 2 1 104 Algebra M07_CME_SB_IGCSE_6867_U07.indd 104 28/8/09 11:35:28 Expanding brackets and factorising EXAMPLE 5 Expand these expressions. 2 1 2 2 (a) 2(3t 4) (b) 3(4x 1) 22 × 3 5 26 22 × 4 5 28 (a) 22(3t 1 4) 5 22 × 3t 1 22 × 4 5 26t 1 28 5 26t 2 8 (b) 23(4x 2 1) 5 23 3 4x 1 23 3 21 23 × 4 5 212 23 × 21 5 13 5 212x 1 3 EXAMPLE 6 Expand then simplify these expressions. (a) 3(2t 1 1) 2 2(2t 1 4) (b) 8(x 1 1) 2 3(2x 2 5) (a) 3(2t 1 1) 2 2(2t 1 4) 5 6t 1 3 2 4t 2 8 Remember to multiply both terms 5 6t 2 4t 1 3 2 8 in the second bracket by 22. 5 2t 2 5 1 2 2 5 1 2 1 (b) 8(x 1) 3(2x 5) 8x 8 6x 15 Expand the brackets first. 5 8x 2 6x 1 8 1 15 Remember that 23 × 25 5 115. 5 2x 1 23 Then collect like terms. EXERCISE 7C Expand these expressions. 1 22(2k 1 4) 2 23(2x 1 6) 3 25(3n 2 1) 4 24(3t 1 5) 5 23(4p 2 1) 6 22(3x 2 7) Expand then simplify these expressions. 7 3(y 1 4) 1 2y 1 10 8 2(k 1 6) 1 3k 1 9 9 4(a 1 3) 2 2a 1 6 10 3(t 2 2) 1 4t 2 10 11 3(2y 1 3) 1 2(y 1 5) 12 4(x 1 7) 1 3(x 1 4) 13 3(2x 1 5) 1 2(x 2 4) 14 2(4n 1 5) 1 5(n 2 3) 15 3(x 2 5) 1 2(x 2 3) 16 4(2x 2 1) 1 2(3x 2 2) 17 3(2b 1 1) 2 2(2b 1 4) 18 4(2m 1 3) 2 2(2m 1 5) 19 2(5t 1 3) 2 2(3t 1 1) 20 5(2k 1 2) 2 4(2k 1 6) 21 8(a 1 1) 2 3(2a 2 5) 22 2(4p 1 1) 2 4(p 2 3) 23 5(2g 2 4) 2 2(4g 2 6) 24 2(w 2 4) 2 3(2w 2 1) 25 x(x 1 3) 1 4(x 1 2) 26 x(2x 1 1) 2 3(x 2 4) Algebra 105 M07_CME_SB_IGCSE_6867_U07.indd 105 28/8/09 11:35:29 7.2 Solving equations involving brackets Equations sometimes involve brackets.